Abstract
A variational Poisson structure on a differential equation \(\mathcal {E}\) is a \(\mathcal {C}\)-differential operator that takes cosymmetries of \(\mathcal {E}\) to its symmetries and possesses the necessary integrability properties. In the literature on integrable systems, Poisson structures are traditionally called Hamiltonian operators. We expose here the computational theory of local variational Poisson structures for normal equations. In this chapter the solutions of Problems 1.24, 1.25, 1.26, and 1.28 is presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Blaszak, M., Szablikowski, B.M.: Classical r-matrix theory of dispersionless systems: Ii. (2 + 1)-dimension theory. J. Phys. A 35, 10345 (2002). arXiv:nlin/0211018
Dubrovin, B.A., Novikov, S.P.: Poisson brackets of hydrodynamic type. Soviet Math. Dokl. 30, 651–654 (1984)
Dubrovin, B.A.: Geometry of 2d topological field theories. Lect. Notes Math. 1620, 120–348 (1996)
Ferapontov, E.V., Galvao, C.A.P., Mokhov, O., Nutku, Y.: Bi-hamiltonian structure of equations of associativity in 2-d topological field theory. Commun. Math. Phys. 186, 649–669 (1997)
Ferapontov, E.V., Lorenzoni, P., Savoldi, A.: Hamiltonian operators of Dubrovin–Novikov type in 2d. Lett. Math. Phys. 105(3), 341–377 (2014). arXiv:1312.0475
Ferapontov, E.V., Pavlov, M.V., Vitolo, R.F.: Projective-geometric aspects of homogeneous third-order hamiltonian operators. J. Geom. Phys. 85, 16–28 (2014). https://doi.org/10.1016/j.geomphys.2014.05.027
Ferapontov, E.V., Pavlov, M.V., Vitolo, R.F.: Towards the classification of homogeneous third-order Hamiltonian operators. Int. Math. Res. Not. 22, 6829–6855 (2016)
Ferapontov, E.V., Pavlov, M.V., Vitolo, R.F.: Systems of conservation laws with third-order Hamiltonian structures’, to appear in Lett. Math. Phys. (2018)
Kersten, P., Krasil′shchik, I., Verbovetsky, A.: On the integrability conditions for some structures related to evolution differential equations. Acta Appl. Math. 83, 167–173 (2004)
Krasil′shchik, J., Verbovetsky, A.M.: Geometry of jet spaces and integrable systems. J. Geom. Phys. 61, 1633–1674 (2011). arXiv:1002.0077
Neyzi, F., Nutku, Y., Sheftel, M.B.: Multi-hamiltonian structure of Plebanski’s second heavenly equation. J. Phys. A 38, 8473 (2005)
Pavlov, M.V., Vitolo, R.F.: On the bi-Hamiltonian geometry of the WDVV equations. Lett. Math. Phys. 105(8), 1135–1163 (2015)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Krasil’shchik, J., Verbovetsky, A., Vitolo, R. (2017). Variational Poisson Structures. In: The Symbolic Computation of Integrability Structures for Partial Differential Equations. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-71655-8_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-71655-8_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71654-1
Online ISBN: 978-3-319-71655-8
eBook Packages: Computer ScienceComputer Science (R0)